Hyper–(Abelian–by–finite) groups with many subgroups of finite depth
Annales mathématiques Blaise Pascal, Volume 14 (2007) no. 1, pp. 17-28.

The main result of this note is that a finitely generated hyper-(Abelian-by-finite) group $G$ is finite-by-nilpotent if and only if every infinite subset contains two distinct elements $x$, $y$ such that ${\gamma }_{n}\left(〈x\text{,}\phantom{\rule{4pt}{0ex}}{x}^{y}〉\right)$ $={\gamma }_{n+1}\left(〈x\text{,}\phantom{\rule{4pt}{0ex}}{x}^{y}〉\right)$ for some positive integer $n=n\left(x,y\right)$ (respectively, $〈x,{x}^{y}〉$ is an extension of a group satisfying the minimal condition on normal subgroups by an Engel group).

Le principal résultat de cet article est qu’un groupe $G$ hyper-(Abélien-par-fini) de type fini est fini-par-nilpotent si, et seulement si, toute partie infinie de $G$ contient deux éléments distincts $x,y$ tels que ${\gamma }_{n}\left(〈x,{x}^{y}〉\right)={\gamma }_{n+1}\left(〈x,{x}^{y}〉\right)$ pour un certain entier positif $n=n\left(x,y\right)$ (respectivement, $〈x,{x}^{y}〉$ est une extension d’un groupe vérifiant la condition minimale sur les sous-groupes normaux par un groupe d’Engel).

DOI: 10.5802/ambp.224
Classification: 20F22, 20F99
Keywords: Infinite subsets, finite depth, Engel groups, minimal condition on normal subgroups, finite-by-nilpotent groups, finitely generated hyper-(Abelian-by-finite) groups
Mot clés : Parties infinies, profondeur finie, Les groupes d’Engel, La condition minimale sur les sous-groupes normaux, les groupes fini-par-nilpotents, les groupes hyper-(Abelien-par-fini) de type fini
Fares Gherbi 1; Tarek Rouabhi 1

1 Department of Mathematics Faculty of Sciences Ferhat Abbas University Setif, 19000 ALGERIA
@article{AMBP_2007__14_1_17_0,
author = {Fares Gherbi and Tarek Rouabhi},
title = {Hyper{\textendash}(Abelian{\textendash}by{\textendash}finite) groups with many subgroups of finite depth},
journal = {Annales math\'ematiques Blaise Pascal},
pages = {17--28},
publisher = {Annales math\'ematiques Blaise Pascal},
volume = {14},
number = {1},
year = {2007},
doi = {10.5802/ambp.224},
zbl = {1131.20024},
language = {en},
url = {https://ambp.centre-mersenne.org/articles/10.5802/ambp.224/}
}
TY  - JOUR
AU  - Fares Gherbi
AU  - Tarek Rouabhi
TI  - Hyper–(Abelian–by–finite) groups with many subgroups of finite depth
JO  - Annales mathématiques Blaise Pascal
PY  - 2007
SP  - 17
EP  - 28
VL  - 14
IS  - 1
PB  - Annales mathématiques Blaise Pascal
UR  - https://ambp.centre-mersenne.org/articles/10.5802/ambp.224/
DO  - 10.5802/ambp.224
LA  - en
ID  - AMBP_2007__14_1_17_0
ER  - 
%0 Journal Article
%A Fares Gherbi
%A Tarek Rouabhi
%T Hyper–(Abelian–by–finite) groups with many subgroups of finite depth
%J Annales mathématiques Blaise Pascal
%D 2007
%P 17-28
%V 14
%N 1
%I Annales mathématiques Blaise Pascal
%U https://ambp.centre-mersenne.org/articles/10.5802/ambp.224/
%R 10.5802/ambp.224
%G en
%F AMBP_2007__14_1_17_0
Fares Gherbi; Tarek Rouabhi. Hyper–(Abelian–by–finite) groups with many subgroups of finite depth. Annales mathématiques Blaise Pascal, Volume 14 (2007) no. 1, pp. 17-28. doi : 10.5802/ambp.224. https://ambp.centre-mersenne.org/articles/10.5802/ambp.224/

[1] A. Abdollahi Finitely generated soluble groups with an Engel condition on infinite subsets, Rend. Sem. Mat. Univ. Padova, Volume 103 (2000), pp. 47-49 | Numdam | MR | Zbl

[2] A. Abdollahi Some Engel conditions on infinite subsets of certain groups, Bull. Austral. Math. Soc., Volume 62 (2000), pp. 141-148 | DOI | MR | Zbl

[3] A. Abdollahi; B. Taeri A condition on finitely generated soluble groups, Comm. Algebra, Volume 27 (1999), pp. 5633-5638 | DOI | MR | Zbl

[4] A. Abdollahi; N. Trabelsi Quelques extensions d’un problème de Paul Erdos sur les groupes, Bull. Belg. Math. Soc., Volume 9 (2002), pp. 205-215 | MR | Zbl

[5] A. Boukaroura Characterisation of finitely generated finite-by-nilpotent groups, Rend. Sem. Mat. Univ. Padova, Volume 111 (2004), pp. 119-126 | Numdam | MR | Zbl

[6] C. Delizia; A. H. Rhemtulla; H. Smith Locally graded groups with a nilpotence condition on infinite subsets, J. Austral. Math. Soc. (series A), Volume 69 (2000), pp. 415-420 | DOI | MR | Zbl

[7] G. Endimioni Groups covered by finitely many nilpotent subgroups, Bull. Austral. Math. Soc., Volume 50 (1994), pp. 459-464 | DOI | MR | Zbl

[8] G. Endimioni Groups in which certain equations have many solutions, Rend. Sem. Mat. Univ. Padova, Volume 106 (2001), pp. 77-82 | Numdam | MR | Zbl

[9] E. S. Golod Some problems of Burnside type, Amer. Math. Soc. Transl. Ser. 2, Volume 84 (1969), pp. 83-88 | MR | Zbl

[10] P. Hall Finite-by-nilpotent groups, Proc. Cambridge Philos. Soc., Volume 52 (1956), pp. 611-616 | DOI | MR | Zbl

[11] J. C. Lennox Finitely generated soluble groups in which all subgroups have finite lower central depth, Bull. London Math. Soc., Volume 7 (1975), pp. 273-278 | DOI | MR | Zbl

[12] J. C. Lennox Lower central depth in finitely generated soluble-by-finite groups, Glasgow Math. J., Volume 19 (1978), pp. 153-154 | DOI | MR | Zbl

[13] J. C. Lennox; J. Wiegold Extensions of a problem of Paul Erdos on groups, J. Austral. Math. Soc. Ser. A, Volume 31 (1981), pp. 459-463 | DOI | MR | Zbl

[14] P. Longobardi On locally graded groups with an Engel condition on infinite subsets, Arch. Math., Volume 76 (2001), pp. 88-90 | DOI | MR | Zbl

[15] P. Longobardi; M. Maj Finitely generated soluble groups with an Engel condition on infinite subsets, Rend. Sem. Mat. Univ. Padova, Volume 89 (1993), pp. 97-102 | Numdam | MR | Zbl

[16] B. H. Neumann A problem of Paul Erdos on groups, J. Austral. Math. Soc. ser. A, Volume 21 (1976), pp. 467-472 | DOI | MR | Zbl

[17] D. J. S. Robinson Finiteness conditions and generalized soluble groups, Springer-Verlag, Berlin, Heidelberg, New York, 1972 | Zbl

[18] D. J. S. Robinson A course in the theory of groups, Springer-Verlag, Berlin, Heidelberg, New York, 1982 | MR | Zbl

[19] D. Segal A residual property of finitely generated abelian by nilpotent groups, J. Algebra, Volume 32 (1974), pp. 389-399 | DOI | MR | Zbl

[20] D. Segal Polycyclic groups, Cambridge University Press, Cambridge, London, New York, New Rochelle, Melbourne, Sydney, 1984 | MR | Zbl

[21] B. Taeri A question of P. Erdos and nilpotent-by-finite groups, Bull. Austral. Math. Soc., Volume 64 (2001), pp. 245-254 | DOI | MR | Zbl

[22] N. Trabelsi Finitely generated soluble groups with a condition on infinite subsets, Algebra Colloq., Volume 9 (2002), pp. 427-432 | MR | Zbl

[23] N. Trabelsi Soluble groups with many 2-generator torsion-by-nilpotent subgroups, Publ. Math. Debrecen, Volume 67/1-2 (2005), pp. 93-102 | MR | Zbl

Cited by Sources: